Sampling Based On Stochastic Optimization

Sampling method has many applications, so sampling theory attracts muchattention in Statistics. Especially, when the probability density function of adistribution is given, there are many existing sampling methods, such as Mixturemethod, Rejection method, MCMC and Quantile method. However, in literature,research is limited to individual descriptions of instructions about algorithms forimplementation of these methods, while systematic comparison among them hasnot been extensively investigated.Based on the concept of kernel density estimation, this paper firstly definesan"L_2^-distance", which can describe the difference between the probability den-sity function of the distribution and kernel density estimation of a sample. Thenaiming to minimize"L_2^-distance", a stochastic optimization sampling algorithmis proposed accordingly. Finally, various sampling methods including Mixturemethod, Rejection method, MCMC, Quantile method and Stochastic optimiza-tion methods are compared when a distribution with complex probability densityfunction is considered. Experiments with different sample size are conducted.Means and standard deviations of L_2^-distances are calculated. Time elapsed andk-th moments of different sampling methods are recorded. All results show thatthough a little more time may be need, the stochastic optimization algorithmproposed in the current paper performs better related to stability and accuracy.It generally outperforms other existing sampling methods in comprehensive eval-uation.